We note that, $L^{2}(\mathbb R)$ is not closed under point wise multiplication.

Let $s>\frac{1}{2};$ and we define Sobolev space, as follows: $H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$

My Question is: Is Sobolev space $H^{s}(\mathbb R),$ for $s>\frac{1}{2},$ closed under point wise multiplication, that is, if $f,g \in H^{s}(\mathbb R),$ can we expect $fg\in H^{s}(\mathbb R)$ ?

Thanks,